Page 5 - Real Analysis I-e content (1)_Neat

Some properties of Supremum

Theorem: Approximation property

Let S be a non empty set of real numbers with a supremem, say b  sup S . Then for every
a  b there is some x in S such that

a  x  b

Proof:

First of all, x  b x in S

If we had x  a for every x in S , then a would be an upper bound for S small than the
least upper bound.

 x  a for atleast one x in S .

 a  x

We have a  x  b.

Thus the theorem is proved.

Theorem: Additive property

Given non empty subsets A and B of R , let C denote the set C  x  y; x A, y B 


If each A and B has a supremum then C has a supremum and sup C  sup A sup B
Proof:

Let a  sup A

b  sup B

If z  C then z  x  y , where x  A, y  B

So z  x  y  a  b

b
Hence a  is an upper bound for C
 C has a supremum, say c  sup C & c  a  b

Next we show that

a  b  c

Choose  0 by approximation property we have